This example shows how a medieval war machine can be optimized by including the elasticity of parts in the dynamical simulation.
A trebuchet is a type of catapult, which was used to throw projectiles at enemies during sieges in the middle ages. The trebuchet makes use of the mechanical advantage of a lever, using the energy from a falling hinged counterweight to launch a payload. Since the counterweight is closer to the pivot than the end with the sling, the payload will achieve higher speed than the counterweight. However, in order to achieve a high launch velocity, the weight of the counterweight must be much greater than the weight of the payload.
The motion of a trebuchet during launch is quite complicated, but accurate predictions can be made using analytical expressions. However, these expressions normally assume that the arm is rigid, as the bending stiffness is difficult to include. Dynamic finite element analyses are therefore needed to accurately predict the trajectory of the payload with different arm stiffness.
Figure: Typical trebuchet design (adapted from ).
In his paper “Trebuchet Mechanics” , Donald B. Siano used the results from thousands of simulations to find the optimal trebuchet design with regards to “range efficiency”, defined as the ratio of the range attained by the trebuchet to the maximum theoretical range. The following conclusions were made:
• The arm should have an initial angle of 45° with respect to the vertical.
• The length of the sling should be equal to the length of the long arm (the payload side of the pivot).
• The long arm should be four times longer than the short arm.
• The counterweight/payload weight ratio should be about 100.
The trebuchet is modelled in FEDEM using beam elements. The frame is fixed to ground, and a revolute joint is attached to the arm and frame to allow rotation of the arm. The counter-weight and payload are modelled as rigid parts with adjustable mass, connected to the arm using stiff axial springs. The height of the frame is 5.0 m. The counterweight and payload have masses of 1000 kg and 10 kg, respectively.
Figure. FEDEM model
The challenging part is to model the interaction between the pouch and the payload. The payload should be lying inside the pouch until the ring slips of the finger, and the payload is released. In the FEDEM model, this feature is simplified using a joint and a simple control system. The joint, which has a high stiffness in all degrees of freedom, is attached to the payload and the sling. The control system measures the rotation angle of the arm, and sets the joint stiffness to zero when the revolute joint reaches a predefined angle. This allows the payload to move without any constraints.
The graph below shows the path of the payload in the simulations with different arm stiffness, where optimal stiffness is defined as the stiffness giving maximum payload range. The optimal bending stiffness, EI, was found to be 5.25e6 Nm2.
Using an arm with 10 % of the optimal bending stiffness introduces significant local dynamics near the sling, which is seen to reduce the range substantially. The results also show that a range increase of 9 % is achieved if an arm with optimal stiffness is used instead of a rigid one.
 F. Normani, "http://www.real-world-physics-problems.com/trebuchet-physics.html," [Online].
 D. B. Siano, "Trebuchet Mechanics," March, 2001.